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Map F Goes from manifold MR All points in M go to R Smooth, i.e. differentiable Function f(x, y) Map UR Open set UM Region U is diffeomorphic to E2 (or En) Scalar Field
One-Form • The scalar field F is differentiable. • Expressed in local variables • f associated with a chart • Need knowledge of the coordinates • from the local chart f • short cut is to use F. • The entity dF is an example of a one-form.
The derivative of the one-form can be written as an operator. Chain rule applied to x, y A point can be described with other coordinates. Partial derivatives affected by chain rule Write with constants reflecting a transformation Operator and Coordinates
The partial derivatives point along coordinate lines. Not the same as the coordinates. Partial Derivative y y = const. x x = const. Y Y = const. X = const. X
Vector Field • General form of differential operator: • Smooth functions A, B • Independent of coordinate • Different functions a, b • Transition between charts • This operator is a vector field. • Acts on a scalar field • Measures change in a direction F(p’) x p’ F(p) p
Inner Product • The one-form carries information about a scalar field. • Components for the terms • The vector field describes how a scalar field changes. • The inner product gives a specific scalar value. • Express with components • Or without
Three Laws • Associativity of addition • Associativity of multiplication • Identity of a constant k = const.
Vector field on Q Contravariant vectors Components with superscripts Transformation rule: One form on Q Covariant vectors Components with subscripts Dual Spaces next